Systems of Linear Equations
Part 2: Multiple Solutions
In this section we will see a different kind
of result from that in Section 1.
- In your workspace,
construct the coefficient
matrix, the vector of constants, and the augmented matrix for the following
system.
2x1 + 4x2 - 4x3 + 4x4
+ x5 = 7
x1 + 2x2 - 2x3 + 2x4
- 2x5 = 1
x1 + 2x2 - 2x3 + 2x4
+ 3x5 = 6
x1 + x2 - 3x3 + 2x4
+ x5 = 2
- Solve the system,
and express your solutions
for each of the x's clearly and carefully.
- Now answer the following
questions:
- How many
free parameters appear in your solution set?
- How many non-zero
equations were in the reduced system?
- How many unknowns
were in the original system?
- These questions are
meant to suggest an important
relationship. To see another example, solve the following system,
and then answer the same questions for this system.
x1 - 2x2 - x3 + 3x4 = 7
-x1 + 2x2 + 3x3 - 3x4 = -3
2x1 - 4x2 - 2x3 + 7x4 = 15
x1 - 2x2 - x3 + 3x4 = 7
- How many free
parameters appear in your solution set?
- How many
non-zero equations were in the reduced system?
- How many unknowns
were in the original system?
- At this point you should
be able to articulate
the relationship between the number of free variables, the number
of non-zero equations in the reduced row echelon system, and the
number of unknowns -- at least for a system with an infinite
number of solutions.
Describe that relationship in your worksheet.
- If your answer to the
challenge in the preceding
paragraph is correct, then you will see that the solution for the first
system in Part 1 (where we found exactly one solution) is consistent with
your answer. Explain why the result from the system in Part 1 is consistent
with your answer, or correct your answer so that the results from all the
systems solved so far fit your answer.
- What does your observation
from the last paragraph
imply about the number of solutions you could find in a given
system?
For example, could there be exactly two solutions?
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